a complexity model for assembly supply chains and its...

A Complexity Model for Assembly Supply Chains andIts Application to Configuration Design

Hui Wang ∗

Department of Industrial and Operations EngineeringUniversity of Michigan, Ann Arbor, MI 48109

Jeonghan KoDepartment of Industrial and Management Systems Engineering

University of Nebraska-Lincoln, Lincoln, NE, 68588

Xiaowei Zhu, S. Jack HuDepartment of Mechanical Engineering

University of Michigan, Ann Arbor, MI 48109

Abstract

A complexity measure for assembly supply chains has been proposed based on

Shannon’s information entropy. This paper extends the definition of such a measure

by incorporating the detailed information of the supply chain structure, the number of

variants offered by each node in the supply chain, and the mix ratios of the variants

at each node. The complexity measure is then applied to finding the optimal assembly

supply chain configuration given the number of variants offered at the final assembler

and the mix ratios of these variants. The optimal assembly supply chain configuration

is theoretically studied in two special scenarios: 1) there is only one dominant variant

among all the variants offered by the final assembler, and 2) demand shares are equal

across all variants at the final assembler. It is shown that in the first scenario where

∗Corresponding author; Address: Department of Industrial and Operations Engineering, University ofMichigan, 1205 Beal Avenue, Ann Arbor, MI 48109, USA; phone: 734-764-9103; e-mail:[email protected]

1

one variant dominates the demand, the optimal assembly supply chain should be non-

modular; but in the scenario of equal demand shares, a modular supply chain is better

than non-modular one when the product variety is high. Finally a methodology is

developed to find the optimal supply chain with/without assembly sequence constraints

for general demands.

1 Introductions

The product variety offered by many industries has increased greatly in the past few decades.

For example, the number of distinct vehicle models in the US increased from 44 in 1969

to 165 in 2005 [1,2]. Another example is the number of styles of running shoes, which

increased from 5 in early 70s to 285 in the late 90s [3]. Such increases were motivated

by the desire to provide high product variety and highly customized products in response

to fierce competition in the market. However, high variety brings a lot of challenges to

manufacturing. Several studies have shown that high product variety has negative impact on

manufacturing system performance, such as increasing manufacturing complexity, degrading

quality, lowering productivity [4,5].

In order to cope with the challenges caused by high variety, manufacturers implement

modular designs in their products. A modular design decomposes the product into several

different modules with standard interface. High product variety is achieved through the

combinational assembly of different modules. Each module still keeps the high production

volume, through which the economy of scale can be maintained.

An important trend enabled by modular products is that manufacturers move from non-

Hui Wang MANU-09-110 2

Figure 1: Manufacturers move from non-modular assembly supply chains to modular assem-

bly supply chains

modular assembly supply chains to modular ones, as shown in Figure 1. In a modular assem-

bly supply chain, the manufacturer apportions the product into different sub-assemblies, each

of which is obtained through the assembly of several modules. Some of the sub-assemblies are

outsourced to the suppliers and the manufacturer only does the final assembly of a few sub-

assemblies. For example, Volvo’s S80 model is assembled from 23 different sub-assemblies,

delivered to the final assembly line through 17 pre-assembled units, 11 of which are operated

by the suppliers [6].

When a manufacturer wants to introduce a new modular product into the market, there

are two important decisions that need to be made in the design stage: 1) how to apportion

the product into different modules; 2) how many variants to offer in each module. These

two decisions will then determine product variety, i.e., the number of variants in the final

assembled product, offered by the manufacturer to customers. Based on that product variety,

the manufacturing system layout and assembly supply chain configuration can be selected

by analyzing and comparing the cost of different systems and supply chain configurations.

However, cost analysis of assembly supply chains requires the estimation of many parameters,


such as manufacturing cost, overage and shortage cost, transportation and production lead

time, etc. Some of these parameters are difficult to estimate in practice. In addition,

sophisticated cost models bring a lot of analytical challenges due to the network structure of

an assembly supply chain, especially when the product variety is taken into consideration.

That is why most research is based on empirical case studies when studying the effect of

product variety on supply chain configurations [7,8].

A new complexity measure for assembly supply chains has been proposed recently by

Wang et al. [9]. This complexity measure is based on Shannon’s information entropy and

takes the following factors into consideration: the supply chain structure, product variety

level of each node and the mix ratios of variants offered by each node. In addition, Wang et

al. [9] investigated the relationship between the complexity and cost of an assembly supply

chain and showed that the cost and complexity agree with each other when comparing

assembly supply chains with the same level of product variety, but different configurations.

It is shown that the decisions made by the complexity and cost criterion follow the same trend

and more specifically, both complexity and cost, based on configuration selection, shift their

preference from non-modular assembly supply chains to modular assembly supply chains, as

product variety increases.

This paper extends the complexity model to incorporate more detailed information on

supply chain structure and product variety. Then the complexity model is applied to the

selection of optimal assembly supply chain configuration, given the number of variants of-

fered by the manufacturer (the final assembler in the assembly supply chain). The paper is

organized as follows. In Section 2, we review the relevant literature. In Section 3, we derive

the complexity model based on Shannon’s information entropy by incorporating detailed


information on supply chain structure and product variety. In Section 4, we theoretically

investigate the supply chain configuration selection problem in two special scenarios. In

Section 5, we study the supply chain selection problem in general demand scenario. We

first develop a decomposition iterative algorithm to generate all possible supply chain candi-

dates without assembly sequence constraints. Then, we extend the results and develop the

methodologies to find the optimal assembly supply chain when product assembly sequence

constraints exist. Section 6 gives some further discussion about this complexity research and

Section 7 concludes the paper.

2 Literature Review

Assembly supply chain is an important research area in supply chain management and most

research in this area is based on inventory management and cost analysis. Early research

of assembly supply chains focuses on centralized assembly systems, where a single decision-

maker determines the ordering policies of all members in the supply chain to minimize the

total cost. Schmidt and Nahmias [10] investigated a finite-horizon model of a centralized

assembly supply chain with two components. Rosling [11] investigated the periodic review,

infinite-horizon inventory problem in assembly supply chains under random demands. Re-

cently decentralized assembly supply chains have gained more and more attention from

researchers. In a decentralized assembly supply chain, each member in the supply chain

makes the decision based on the decisions made by its upstream suppliers and downstream

assembler to minimize its own cost. Wang and Gerchak [12] studied a decentralized assem-

bly supply chain with perfect component yield and stochastic demand for the end product,


where the suppliers make capacity decisions for the production of components. Bernstein and

DeCroix [13] studied equilibrium price and capacity decisions in a multiple-echelon modular

assembly supply chain and showed that the modular assembly supply chain could be more

beneficial than the non-modular assembly supply chain when the introduced sub-assemblers

have lower assembly cost than the final assembler. However, most research in this area fo-

cuses on assembly supply chains of single product. In this paper, we incorporate product

variety into the assembly supply chain model and investigate assembly supply chains using

a new performance measure, complexity.

A few papers address how product variety influences supply chain configuration decision.

Randall and Ulrich [7] used data from U.S. bicycle industry to examine the relationship

among product variety, supply chain structure and system performance. It was shown that

firms that match their supply chain structure to the type of variety they offer often outper-

form those that fail to make such choices. Salvador et al. [8] used empirical data to explore

how a firm’s supply chain, defined as the whole of its supply, manufacturing and distribution

networks, should be configured, when different degrees of customization are offered. We

focus on a similar problem in the context of assembly supply chains, but instead of applying

empirical case study, we use a model-based method to find the solution.

Several different definitions of complexity have been proposed by researchers within dif-

ferent disciplines. For example, Suh [14] defined a complexity measure particularly appli-

cable in product design; Cover and Thomas [15] discussed Kolmogorov complexity, which

is a measure of computational resources needed to describe a string of text. A commonly-

used complexity definition is based on the information entropy, proposed by Shannon [16]

in the context of communications systems. Shannon’s information entropy is a measure of


the uncertainty surrounding the outcome of a random experiment. Suppose we have a ran-

dom experiment with n possible outcomes, whose probabilities of occurrence are q1, q2, ..., qn.

Then the information entropy of this random experiment, showing how uncertain we are of

the outcome, is of the following form,

H = −Kn∑

i=1

qi log2 qi (1)

where the positive constant K merely amounts to a scaling factor.

Information entropy plays an important role in communication systems and other dis-

ciplines as a measure of information, choice and uncertainty. It has been used to study

complexity in many different areas, including biology, physics, manufacturing systems, sup-

ply chains, etc [17,18]. For example, Deshmukh et al. [19] derived an information-theoretic

entropy measure of complexity for a given combination and ratio of part types to be produced

in a manufacturing system. Zhu et al. [20] studied the operator choice complexity in mixed-

model assembly lines and developed a methodology to find the optimal assembly sequence

to minimize manufacturing complexity. Sivadasan et al. [21] developed an experimental

methodology to study the operational complexity in a supplier-customer system. Following

the similar method, Frizelle [22] developed a metric to measure the complexity within the

supply chain, called operational complexity. Wang et al. [9] proposed a complexity measure

for assembly supply chains in the presence of product variety and studied the relationship

between complexity and cost of an assembly supply chain. Based on the complexity model,

Wang et al. [23] studied how to make the decision of assembly supply chain configuration

under different product variety level. In this paper we extend the complexity model of Wang

et al. [9] by incorporating more detailed information on supply chain structure and product


variety. Based on this complexity measure, we study the assembly supply chain selection

problem given the product variety.

3 Complexity Model of Assembly Supply Chains

In this section, we derive the complexity model based on Shannon’s information entropy

by incorporating the detailed information on the supply chain structure, the number of

variants each node produces and the mix ratios of the variants offered by each node. Here

we follow the traditional definition of an assembly supply chain (also called assembly system

in supply chain literature) in the context of supply chain management, where each node can

have multiple upstream suppliers, but a given node can only supply one downstream node.

Suppose a general assembly supply chain takes the form in Figure 2. The final product

is apportioned into m different modules and each module has several different variants.

There are n nodes in the supply chain and each node can produce one product (either one

module, such as node 1 and 2, or one sub-assembly from the assembly of several modules,

such as node m + 1) with certain number of variants. We assume that a downstream node

can assemble any combination of the variants provided by its upstream suppliers, and each

combination counts as a distinct variant. This assumption can easily be relaxed by setting

the demand of the non-existing variants as zero. Suppose only one upstream node provides

all the variants of one particular component for its downstream assembler. Following this

assumption, we know that the number of nodes in the most upstream echelon is equal to

the number of modules in the final product, m. Here we number the nodes in the following

sequence by convention. Node 1, 2, . . . ,m are the nodes in the most upstream echelon. Node


Figure 2: A general assembly supply chain and relationships of variants’ demand share

m + 1, . . . , n − 1 are intermediate sub-assemblers and node n is the final assembler. Since

the nodes in the most upstream echelon do not have suppliers and we want to capture all

the supply-assembly activities in the whole supply chain, a virtual supplier is introduced

here, denoted as node 0, which supplies all the raw materials to nodes in the most upstream

echelon. Notice that this virtual supplier, node 0, is a special node in the supply chain

because it has multiple downstream nodes.

As regards the number of variants and their mix ratios, we define the following notations:

Vi : the number of variants that node i can produce, i = 1, . . . , n.

Si : the set of nodes that are suppliers to node i, i = 1, . . . , n.

Aiju : the set of variants produced at node i that use variant u from node j, where node

j is a supplier to node i, i.e., j ∈ Si.

At each time period, one unit product is demanded at node i = 1, . . . , n − 1 from its

downstream assembler and for the final assembler, one unit product is demanded by the


customer. Assume the demands of the variants at node i are independent, i = 1, . . . , n. Let

qiv denote the probability that at each time period, the variant v = 1, . . . , Vi is demanded

at node i, which is also equal to the fraction of node-i demand that belongs to the variant

v in the long run. Hereafter, we refer to qiv as the demand share of variant v at node i.

In addition, define the vector qi := (qi1, qi2, . . . , qi,Vi), which captures the mix ratios of the

variants produced by node i. Hereafter, we refer to qi as the demand vector of node i.

The final assembler’s demand vector, qn, determines how the demand at upstream nodes

is allocated among the variants produced by those nodes (see node 1 and 2 and m + 1 in

Figure 2 for an illustration). Using the notation introduced so far, we have the following

relationship between the demand share of variant u at node j, qju, and the demand vector

of node i, qi, where node j is a supplier to node i, (i.e., j ∈ Si):

qju =∑

v∈Aiju

qiv, j ∈ Si. (2)

The complexity of an assembly supply chain is caused by the following factors: the

supply chain structure, product variety level of each node in the supply chain, and demand

uncertainty faced by each node. The supply chain structure is determined by the number

of nodes in the supply chain and their supply-assembly relationships. The product variety

level of a node is the number of variants produced by this node. The demand uncertainty

a node faces is decided by the mix ratios of the variants at that node, which tells us the

probability that the next demanded unit will be a certain variant. It follows the analogy of

information entropy concept: the more evenly distributed the variants of a node, the higher

uncertainty of the demand the node faces. Therefore the complexity of an assembly supply

chain is determined by and should increase with the following factors: 1) the number of


nodes in the supply chain and their supply-assembly relationships; 2) the number of variants

produced by each node in the supply chain; 3) the evenness level of demand mix ratio across

all the variants offered by a node in the supply chain.

Following the above argument and the information entropy concept, the complexity model

of an assembly supply chain shown in Figure 2, is developed through the following five steps:

Step 1: The number of nodes in the supply chain and their relationships are represented

by an adjacency matrix,Φ:

Φ =

φ00 φ01 . . . φ0n

φ10 φ11 . . . φ1n

......

. . ....

φn0 φn1 . . . φnn

(n+1)×(n+1)

The number of columns and rows of matrix Φ equals to the total number of nodes in the

supply chain including the virtual supplier, n + 1. With regard to the supply-assembly

relationships, if node j is a supplier of node i (i.e., j ∈ Si), then φji = 1; otherwise, φji = 0,

where i = 0, . . . , n and j = 0, . . . , n.

Step 2: For every supply-assembly relationship, where φji = 1 in matrix Φ, matrix

Qji = ((qjiuv)) =

qji00 . . . qji1Vi

.... . .

...

qjiVj1. . . pjiVjVi

Vj×Vi

is used to represent the number of variants of node j and node i and the mix ratio of the

variants of downstream node i. The number of columns of matrix Qji is the number of

variants produced at node i, Vi and the number of rows is the number of variants offered by


node j, Vj. Here, qjiuv, u = 1, . . . , Vj, v = 1, . . . , Vi is the production volume of variant u that

node j needs to produce for the production of variant v at node i in order to satisfy the final

demand from the customer, assuming that the total demand for all the variants offered by

the final assembler is 1.

Step 3: Every matrix Qji = ((qjiuv)) is normalized by

q̃jiuv =qjiuvK

where K =∑i

∑j

∑u

∑v

qjiuv (3)

Step 4: The assembly supply chain is regarded as a system with the states, whose

occurrence probability is q̃jiuv. Based on Shannon’s information entropy formulation in (1),

we define the complexity contribution of any supply-assembly relationship in the supply

chain (φji = 1 in matrix Φ), as follows:

Cji = −∑u

∑v

q̃jiuv log2 q̃jiuv (4)

Step 5: The complexity of an assembly supply chain is obtained by summing the

complexity contribution of all supply-assembly relationships in the supply chain and takes

the following form:

C =∑i

∑j

Cji (5)

The complexity of an assembly supply chain defined by Equation (5) can be calculated

by a simpler formulation (6), where Li is the number of suppliers of node i = 1, 2, . . . , n, and

K =∑n

i=1 Li is the total number of arcs in the supply chain network,

C = −n∑

i=1

Li

Vi∑v=1

qivK

log2

qivK

(6)

Wang et al. [9] also interprets this complexity measure through the following random

experiment. Suppose for each node, we form a pool of variants produced by that node,


Figure 3: The example used to illustrate the calculation of assembly supply chain complexity

where each variant is represented in a quantity proportional to its demand share. Consider

the random experiment where we first pick an arc of the supply chain at random, and we

then pick one item from the pool of this arc’s end-node. The probability of picking a certain

arc, which connects node i to a supplier node j ∈ Si, and then picking variant v from the

pool of node i is qiv/K (since we pick an arc with probability 1/K and variant v of node i

with probability qiv). Wang et al. [9] showed that the information entropy of this random

experiment is given by Equation (6). Loosely speaking, the complexity measure indicates

the level of uncertainty about the next flow of material that will occur in the supply chain.

An example is given here to illustrate how to calculate the complexity of an assembly

supply chain. Suppose one assembly supply chain takes the form as Figure 3. There are five

nodes in the supply chain, where node 1, 2, 3 are in the most upstream echelon, node 5 is the

final assembler and node 4 is the intermediate sub-assembler. There is one virtual supplier,

node 0, which provides all the raw materials to the nodes in the most upstream echelon.

Each node provides a certain number of variants and assembles all the possible combinational

variants provided its suppliers. Suppose the demand vector of the final assembler is q5 =


(12, 18, 0, 1

4, 0, 0, 1

8, 0). Based on the demand vector of the final assembler, the demand vector

of other nodes in the supply chain can be obtained by Equation (2) and then complexity of

this assembly supply chain can be obtained by the following five steps.

Firstly, matrix Φ is obtained as follows:

Φ =

φ00 φ01 φ02 φ03 φ04 φ05

φ10 φ11 φ12 φ13 φ14 φ05

φ20 φ21 φ22 φ23 φ24 φ25

φ30 φ31 φ32 φ33 φ34 φ35

φ40 φ41 φ42 φ43 φ44 φ45

φ50 φ51 φ52 φ53 φ54 φ55

=

0 1 1 0 1 0

0 0 0 1 0 0

0 0 0 1 0 0

0 0 0 0 0 1

0 0 0 0 0 1

0 0 0 0 0 0

Secondly, for any supply-assembly relationship ( i.e., φji = 1 in matrix Φ), matrix Qji is

developed. For example, we have the following matrix Q45 for φ45 = 1,

Q45 =

12

18

0 0 0 0 0 0

0 0 0 14

0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 18

0

Similarly, we get Q01,Q02,Q03,Q14,Q24,Q35. Thirdly, matrix Qji is normalized by Equa-

tion (3), where K = 7 in this example. For instance, we normalize Q45 and obtain the

following normalized matrix Q̃45,

Q̃45 =

114

156

0 0 0 0 0 0

0 0 0 128

0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 156

0


Following the same procedure, we get other normalized matrixes, Q̃01, Q̃02, Q̃03, Q̃14, Q̃24, Q̃35.

Fourthly, the complexity contribution of each supply-assembly relationship, based on each

normalized matrix Q̃ji, is calculated through Equation (4). In this example, we calculate

the complexity contribution of each relationship as follows: C01 = 0.479, C02 = 0.537, C03 =

0.537, C14 = 0.587, C24 = 0.587, C35 = 0.651, C45 = 0.651. Finally the complexity of the

assembly supply chain is calculated by summing the complexity contribution of all supply-

assembly relationships in the supply chain, i.e, C = C01 +C02 +C03 +C14 +C24 +C35 +C45 =

4.029.

4 Configuration Selection for Assembly Supply Chains

in Two Special Scenarios

In this paper, we apply the complexity measure to solving the following decision-making

problem of assembly supply chain configuration: After the manufacturer (the final assembler)

decides the number of variants offered to the customer and supposedly the demand share of

these variants can be estimated, then how should the assembly supply chain be configured

so that the supply chain complexity is minimized? Specifically, we want to compare all the

possible supply chain configurations in terms of complexity and select the one with the least

complexity value as the optimal supply chain configuration. In this section, we theoretically

investigate this configuration selection problem in the following two special scenarios: 1)

there is one dominant variant among all the variants offered by the final assembler, and

2) demand shares are equal across all variants at the final assembler. In the scenario of


one dominant variant, we show that the optimal supply chain configuration should be non-

modular assembly supply chain if the dominance of that variant is big enough. In the scenario

of equal demand shares, we show that modular assembly supply chains are more beneficial

than non-modular ones when the product variety is high.

Let us consider the first scenario. Suppose we are given the number of variants offered by

the final assembler and the mix ratios of these variants. In addition among all the variants

offered by the final assembler to customers, there is one dominant variant, which is preferred

by most customers and whose demand share is much larger than the demand share of other

variants. The following proposition tells us an important property of complexity measure for

assembly supply chains when the demand share of that dominant variant is big enough and

it provides an useful guideline in how to configure the assembly supply chain in the scenario

of one dominant variant.

Proposition 1. Suppose among all Vn variants offered by the final assembler, there exists one

particularly dominant product, denoted as variant 1 for notation convenience, whose demand

share, qn1, is much bigger than other variants. If the demand share of that dominant variant

increases and approaches 1, i.e, qn1 → 1, then

a.) the complexity of the assembly supply chain in Figure 2 only depends on the supply

chain structure and equals to log2K, where K is the total number of arcs in the supply

chain, including the arcs from the virtual supplier to the nodes in the most upstream echelon;

b.) the optimal assembly supply chain configuration should be non-modular assembly supply

chain.

Proposition 1 tells us that given one dominant variant at the final assembler, if the


demand share of that dominant variant is big, we can compare the complexity of different

supply chain configurations by simply comparing the number of arcs in different supply chains

and then the optimal configuration is just the one with least number of arcs in the supply

chain. It also tells us that in the scenario of one dominant variant, introducing intermediate

sub-assemblers increases the complexity of the assembly supply chain, which implies that

the optimal supply chain configuration should be non-modular.

Now let us consider the second scenario. Suppose the number of variants offered by the

final assembler and the mix ratios of these variants are given. Furthermore, we know that

customers have equal preference to all variants offered by the final assembler, which means

the demand shares are the same across all the variants of the final assembler. Therefore, by

Equation (2), the demand shares are also identical across the variants provided by all other

nodes in the supply chain. The following proposition can provide some useful guidance in

the decision of the supply chain configuration in the scenario of equal demand shares.

Proposition 2. Suppose demand shares are equal across all the variants provided by the

final assembler, i.e., qn = ( 1Vn, . . . , 1

Vn)1×Vn. For ease of exposition, we assume all the nodes

in the most upstream echelon, node 1, . . . ,m, provide the same number of variants, i.e,

V1 = . . . = Vm = V and Vn = V m. Then if the number of variants produced at each node in

the most upstream echelon, V , is high, modular assembly supply chains are more beneficial

than non-modular.

Proposition 2 tells us that in the scenario of equal demand shares, modular assembly

supply chains could be more preferable than non-modular ones if product variety level is

high. But it does not give us specific details about how to configure the modular assembly


supply chain, such as how many number of echelons and intermediate sub-assemblers we

should have, how we should allocate the upstream suppliers to the sub-assemblers, etc. The

following proposition provides some useful tips in comparing different modular assembly

supply chains under the scenario of equal demand shares.

Proposition 3. Suppose the demand shares are equal across all the variants provided by the

final assembler, i.e., qn = ( 1Vn, . . . , 1

Vn)1×Vn. In addition, we assume all the nodes in the most

upstream echelon, node 1, . . . ,m, provide the same number of variants, i.e, V1 = . . . = Vm =

V and Vn = V m.

a.) Consider the following two modular assembly supply chains. Both of them have one

middle intermediate echelon. Supply chain I has c intermediate sub-assemblers, each of

which have d suppliers from the most upstream echelon; supply chain II has d intermediate

sub-assemblers, each of which have c suppliers from the most upstream echelon. When the

number of variants at the node in the most upstream echelon, V , is big, then the supply chain

with larger number of intermediate sub-assemblers has less complexity value than another one,

which means if c ≥ d, then complexity of supply chain I is less than supply chain II.

b.) Consider the following two modular assembly supply chains. Both of them have one

middle intermediate echelon and c intermediate sub-assemblers in the middle echelon. In

supply chain I, every intermediate sub-assembler has the identical number of suppliers from

the most stream echelon, d, i.e, Lm+1 = . . . = Lm+c = d. But in supply chain II, the

intermediate sub-assemblers do not have the same number of suppliers from the most stream

echelon, i.e, Lm+i = d + ei, ei ∈ Z,∑c

i=1 ei = 0, i = 1, . . . , c. Then the complexity of supply

chain I is lower than supply chain II.


Figure 4: Three modular assembly supply chains with 6 nodes in the most upstream echelon

and one intermediate echelon

Given the equal demand shares of variants at the final assembler, suppose we decide

to implement modular assembly supply chains due to high product variety and choose to

have one intermediate echelon for sub-assemblers. Then proposition 3 provides some de-

tailed insights regarding to how to configure the modular assembly supply chain. If all

sub-assemblers in one modular assembly supply chain have the same number of suppliers

from the most upstream echelon, we call it a balanced modular assembly supply chain.

Similarly, an unbalanced modular assembly supply chain is a modular supply chain, whose

sub-assemblers in the middle echelon have different number of suppliers. Proposition 3

tells us under the scenario of equal demand shares, if the number of sub-assemblers in the

intermediate echelon is fixed, complexity of the balanced supply chain is lower than the

corresponding unbalanced ones. But for two balanced modular supply chains with different

number of sub-assemblers in the intermediate echelon, the supply chain with larger num-

ber of sub-assemblers is less complex. Following proposition 3, we can easily compare the

complexity of the three configurations of Figure 4 and obtain the following complexity rela-

tionship, if the demand shares are the same for all the variants offered by the final assembler,

Complexity(I) > Complexity(II) > Complexity(III).


5 Optimal Assembly Supply Chain Selection

In section 4, we discussed how to make the configuration decision of assembly supply chains

in two special scenarios: one dominant variant scenario and equal demand shares scenario.

But very often the customer demand at the final assembler can’t be classified into either

of these two special scenarios. Therefore in this section we study how to find the optimal

assembly supply chain under general demands, if we only know the number of variants

offered at the final assembler and the mix ratios of these variants. We first develop a

decomposition iterative algorithm to generate all possible supply chain candidates without

assembly sequence constraints and the optimal one can be obtained through comparing

the complexity values of these candidates. Then, we extend the results and develop the

methodologies to find the optimal assembly supply chain when product assembly sequence

constraints exist.

5.1 Optimal Assembly Supply Chain Selection without Assembly

Constraints

For easy explanation, a specific example shown in Figure 5, is used to illustrate the pro-

cedure. In this example, a modular product is apportioned into 4 modules, module A, B,

C and D, with three, two, one and two variants respectively. Under the assumption of full

combinational assembly, the final assembler produces 3 × 2 × 1 × 2 = 12 different variants.

Suppose the demand share of these 12 variants is estimated as dj, j = 1, . . . , 12. Based on

these conditions, we want to find the optimal assembly supply chain that has the minimum

complexity. Since one upstream node provides all the variants of one particular module (or


Figure 5: The example used to illustrate the methodology of finding the optimal assembly

supply chain

sub-assembly) for its downstream assembler, then all the assembly supply chain candidates

should have the same number of nodes in the most upstream echelon and the same demand

vector at these nodes. In addition, the number of nodes in the most upstream echelon is

equal to the number of modules in the product. In this specific example, there will be four

nodes in the most upstream echelon shown in Figure 5. Because each node in the supply

chain provides the unique product with several variants, in this section we use the name of

the product produced at a node as the index of that node for ease of exposition, instead of

using number as the index (Section 3). For example, here we name the final assembler in

Figure 5 as node ABCD, instead of node n.

The procedure of finding the optimal assembly supply chain is divided into three steps:

Step I: Enumerate all possible supply chain candidates that have one final assembler in the

last echelon and four suppliers in the most upstream echelon, providing module A, B, C and


Figure 6: There are five configurations to connect the four nodes in the most upstream

echelon to the final assembler

D respectively.

Step II: For each supply chain candidate, based on the demand vector of the final assembler,

qABCD = (d1, d2, . . . , d12), calculate the demand vector of all other nodes in the supply chain

by Equation (2) and substitute them back to Equation (6) to get the supply chain complexity.

Step III: Compare the complexity value of all supply chain candidates and obtain the

optimal assembly supply chain by selecting the one with the minimum complexity.

Among these three steps, the first step, enumerating all possible supply chain candidates,

is the most challenging. It is because for a given number of nodes in the most upstream

echelon, there are many ways to connect these nodes to the final assembler, which results

in many different supply chain configurations. In addition, for each configuration there are

more than one supply chain candidate due to different locations of the nodes in the most

upstream echelon. For instance, in our example here, there are five different supply chain

configurations, all of which have four nodes in the most upstream echelon, as shown in

Figure 6. For each configuration, there are several possible supply chain candidates. For

example, the configuration IV in Figure 6 has three different supply chains due to the location

difference of the nodes in the most upstream echelon, as shown in Figure 7.

In order to enumerate all possible supply chain candidates, considering the configuration


Figure 7: There are more than one supply chain candidates for the same configuration

difference and node location difference, a description of an assembly supply chain containing

only characters of letters, ‘(’ and ‘)’ is proposed here. Webbink and Hu [24] applied a similar

string description to represent the system configurations in the context of manufacturing

system design and developed a decomposition algorithm to generate of manufacturing system

configurations consisted of n workstations based on that description. In this paper, we

used the similar description to represent assembly supply chains. In the description, one

pair of parentheses represents an assembly relationship, including the assembly relationships

between the nodes in the most upstream echelon and the virtual supplier. Starting from

the most upstream echelon of the supply chain and moving forward to the final assembler,

when an assembly relationship is met, a pair of parentheses is added. Repeat this process

until the final assembler is reached. For instance, in Figure 7 (a), starting from the most

upstream echelon, there are four supply-assembly relationships, between node A, B, C, D

and the virtual supplier, which are represent as (A), (B), (C) and (D). Moving forward,

module (A) and (C) are assembled together at node AC and one more parenthesis is added,

through which sub-assembly ((A)(C)) is obtained. Sub-assembly ((B)(D)) is obtained in the

similar way. Moving further, sub-assembly ((A)(C)) and ((B)(D)) are assembled at the final

assembler and one more parenthesis is added. Therefore (((A)(C))((B)(D))) is obtained to


represent the supply chain of Figure 7 (a).

After developing this description method for assembly supply chains, we then develop

an iterative decomposition algorithm, that generates all possible assembly supply chain can-

didates, given the number of nodes in the most upstream echelon. It is composed of the

following three steps.

Step 1: According to the modules in the final product, generate the set of all sub-assemblies

and modules, which could be produced by any node in all possible assembly supply chains.

In the example of Figure 5, based on the four modules in the final product, A, B, C and

D, the following set of sub-assemblies and modules is generated, (ABC), (ABD), (ACD),

(BCD), (AB), (AC), (AD), (BC), (BD), (CD), (A), (B), (C), (D).

Step 2: List all the possible assembly combinations of sub-assemblies and modules gener-

ated from Step 1, through which the final product ABCD can be achieved and then one

more parenthesis is added. For the example of Figure 5, the combinations are ((ABC)(D)),

((ABD)(C)), ((ACD)(B)), ((BCD)(A)), ((AB)(CD)), ((AB)(C)(D)), ((AC)(BD)), ((AC)(B)(D)),

((AD)(BC)), ((AD)(B)(C)), ((BC)(A)(D)), ((BD)(A)(C)), ((CD)(A)(B)) and ((A)(B)(C)(D)).

Step 3: For each assembly combination in step 2, check whether the cardinality of each

inner parentheses is one or not. If the cardinality of an inner parentheses is more than

one, it means there is a sub-assembly in this inner parenthesis. Then that sub-assembly is

treated as the final product in Step 1. Go to step 1 and step 2. Repeat this process until

the cardinalities of all inner parentheses are one, which means no more sub-assembly needs

to be decomposed. For the combination instance of ((ABC)(D)), the cardinality of inner

parentheses (ABC) is three, more than one. Treat sub-assembly (ABC) as the final product.

Redo step 1 and Step 2, in which we get the following set, (AB), (AC), (BC), (A), (B), (C)


Figure 8: Iterative decomposition algorithm to generate all possible supply chain candidates

and × stands for the infeasible candidates, which will be discussed in section 5.2

and possible assembly combinations, ((AB)(C)), ((AC)(B)), ((BC)(A)), ((A)(B)(C)). Take

(((AB)(C)) for an instance and the cardinality of (AB) is more than one. Repeat Step 1 and

Step 2. Only one assembly combination ((A)(B)) is obtained. Since the cardinality of all

inner parentheses in ((A)(B)) is one, we stop. Replace (AB) in combination ((AB)(C)) with

((A)(B)) and then we get (((A)(B))(C)). Following the same procedure, we replace (ABC)

in combination ((ABC)(D)) with (((A)(B))(C)) and finally obtain ((((A)(B))(C))(D)).

By this iterative decomposition algorithm, we can generate all possible supply chain can-

didates, shown in Figure 8. For each candidate, based on the demand vector of the final

assembler, the demand vector of all the nodes in the supply chain is calculated by Equa-

tion (2) and then the corresponding complexity of that assembly supply chain is calculated

through Equation (6). The optimal assembly supply chain is achieved through comparing

complexity of all supply chain candidates and selecting the one with the least complexity.


5.2 Optimal Assembly Supply Chain Selection with Assembly

Constraints

In the previous section, we developed an iterative algorithm to generate all possible supply

chain candidates if the number of variants offered by the final assembler and their mix ratios

are given. In that algorithm, we assume that there is no assembly sequence constraint and

the final product can be obtained through any possible assembly sequence. But in practice,

it may not be true. Assembly sequence constraints often exist in the product assembly

process. For example, a laptop assembly process requires the keyboard to be assembled

after all other components are assembled to the main board. In this section, we investigate

the optimal assembly supply chain with assembly sequence constraints, given the number of

variants at the final assembler and their mix ratios. The same example, shown in Figure 5,

is used again to illustrate the procedure. Besides the knowledge of 12 variants offered by the

final assembler with the demand share, dj, j = 1, . . . , 12, we also have the following assembly

sequence constraints, represented in Figure 9: 1) Module A and B must be assembled before

module C; 2) Module D can not be assembled until module C is assembled. If i � j is used

to represent a constraint that module i must be assembled before module j, then the above

assembly sequence constraints can be written as:

A � C,B � C,C � D (7)

The procedure of finding the optimal supply chain with assembly sequence constraints is

to first generate all feasible supply chain candidates that satisfy all the assembly sequence

constraints and then compare complexity of these feasible supply chain candidates and finally

obtain the optimal supply chain by selecting the one with the least complexity value. One


Figure 9: A set of assembly sequence constraints

straightforward method of obtaining feasible supply chain candidates is to first generate all

the supply chain candidates without assembly constraints by the iterative decomposition

algorithm developed in section 5.1, then check the feasibility of each candidate and delete

the candidates that do not satisfy the assembly constraints. This method can be illustrated

in Figure 8, where × stands for the infeasible supply chain candidates that do not satisfy

constraints (7).

This method is easy to understand and apply, but the efficiency is low because of the

number of supply chain candidates generated under the assumption of no assembly sequence

constraint, which increases exponentially with the increase of the number of modules in the

product. Here, we develop a more efficient method, which generates all feasible supply chain

candidates without generating and checking all possible supply chain candidates. Recall that

in the iterative decomposition algorithm of section 5.1, we keep decomposing the assembly

combinations obtained in step 2 until the cardinality of each inner parenthesis is one. Now,

instead of checking the feasibility after all the decomposition is done, we check the feasibility

of every intermediate supply chain candidate that is obtained after each decomposition.

The infeasible intermediate supply chain candidates are deleted and in next decomposition

iteration, no more decomposition is performed on these infeasible intermediate candidates, as

shown in Figure 10. Compared with the previous method, the number of feasibility check is

reduced, because infeasible intermediate supply chain candidates stop further decomposition


Figure 10: Feasibility check of intermediate supply chain candidates after each decomposition

and no more feasibility check is performed from this point.

Here the example shown in Figure 5, is used to illustrate this method. In this ex-

ample, we want to generate all feasible supply chain candidates that satisfy the following

assembly sequence constraints (7). First, in step 2 of the iterative decomposition algo-

rithm, we obtain the following 14 intermediate supply chain candidates after 1st decom-

position, ((ABC)(D)), ((ABD)(C)), ((ACD)(B)), ((BCD)(A)), ((AB)(CD)), ((AB)(C)(D)),

((AC)(BD)), ((AC)(B)(D)), ((AD)(BC)), ((AD)(B)(C)), ((BC)(A)(D)), ((BD)(A)(C)), ((CD)

(A)(B)) and ((A)(B)(C)(D)). Second, we check the feasibility of these 14 intermediate supply

chain candidates before we start the next decomposition. The intermediate supply chains

that do not satisfy the constraints, are deleted. Third, the 2nd decomposition only starts

with the following three feasible ones, ((ABC)(D)), ((AB)(C)(D)) and ((A)(B)(C)(D)). Then

the same procedure repeats until the last decomposition is done. Figure 10 summaries the

details of this method. This method helps to reduce the number of feasibility check from 26


to 13.

In the iterative decomposition algorithm developed in section 5.1, we start from the fi-

nal product and in each iteration, we decompose the final product (1st iteration) or the

sub-assemblies (remaining iterations) into assembly combinations, which are composed of

smaller-size sub-assemblies and modules. Let P k represents the sub-assembly, which is de-

composed in the kth iteration (notice that P 1 is the final product in the first iteration). Here

we use the following criteria to check whether constraint i � j is satisfied in one intermediate

assembly combination, which is generated through the decomposition of sub-assembly P k.

First, we check whether sub-assembly P k contains both module i and module j. If not, we

conclude that this assembly combination satisfies constraint i � j, because the feasibility

has already been guaranteed in the previous decompositions. If P k contains both module i

and j, then we check whether this assembly combination, generated through the decompo-

sition of P k, satisfies one of the following two requirements: 1) In this decomposition, P k

is decomposed into the combination of module j and other sub-assemblies (or modules); 2)

In this decomposition, P k is decomposed into the combination of a sub-assembly containing

module j and other sub-assemblies (or modules) and this sub-assembly containing module

j also contains module i. If one of the above two requirements is met, then it is concluded

that constraint i � j is satisfied in this intermediate assembly combination; otherwise, we

conclude that constraint i � j is not satisfied and this intermediate assembly combination

is infeasible. It is deleted and no more decomposition is performed on this combination in

the next iteration. Figure 11 summaries the details of this feasibility check criteria.

We use Figure 10 as an example to illustrate this procedure. In first iteration, P 1 =

(ABCD) is decomposed into 14 intermediate assembly combinations. We choose one in-


Figure 11: Feasibility check criteria for an intermediate assembly supply chain

termediate combination, ((ABC)(D)), to check whether it satisfies constraints (7). For

constraint A � C, first we find that P 1 = (ABCD) contains both module A and mod-

ule C. Then the intermediate combination ((ABC)(D)) is obtained through decomposing

P 1 = (ABCD) into a sub-assembly (ABC) that contains module C and another module (D).

The sub-assembly containing module C, (ABC), also contains module A. So we conclude

that assembly combination ((ABC)(D)) satisfies constraint A � C. Similarly, intermediate

combination ((ABC)(D)) also satisfies constraint B � C. For constraint C � D, the in-

termediate combination ((ABC)(D)) is obtained through decomposing P 1 = (ABCD) into

module D and another sub-assembly (ABC), so constraint C � D is also met. Then we

conclude that ((ABC)(D)) is a feasible intermediate supply chain candidate and in next it-

eration, it will be further decomposed. Then in the second iteration, i = 2, the sub-assembly

(ABC), i.e., P 2 = (ABC), is decomposed into the following four intermediate assembly com-

binations ((AB)(C)), ((AC)(B)), ((BC)(A)) and ((A)(B)(C)). Here we select combination

((AC)(B)) as an example to check whether it is a feasible intermediate combination. For

constraint C � D, the sub-assembly decomposed in this iteration, P 2 = (ABC), does not

contain module D. Therefore, we conclude that ((AC)(B)) satisfies constraint C � D. For

constraint B � C, first the sub-assembly decomposed P 2 = (ABC) contains both module


B and module C. Then in this decomposition, the intermediate sub-assembly ((AC)(B)) is

obtained through decomposing P 2 = (ABC) into a sub-assembly (AC) that contains module

C and another module (B). The sub-assembly containing module C, (AC), does not con-

tain module B. Therefore, intermediate assembly combination ((AC)(B)) does not satisfy

constraint A � C and is an infeasible intermediate supply chain. So we delete it and in next

iteration, no further decomposition will performed.

6 Discussions

One challenge of this complexity research is that when the iterative decomposition algorithm

is used to generate possible supply chain candidates, the number of candidates increases

exponentially as the number of modules in the product increases. If the number of modules

is big, it is difficult to generate all possible supply chain candidates, especially when no

assembly constraint exists. As discussed in section 3, if the demand shares of the variants

at the final assembler fall into one of the following two extreme scenarios, one dominant

variant and equal demand shares, the optimal supply chain can be easily obtained without

generating all possible supply chain candidates. But unfortunately, very often the final

assembler’s demand does not fit into these two scenarios and the iterative decomposition

algorithm has to be used. So in these cases, it is very challenging to study the optimal

supply chain problem if the number of modules is high and there is no assembly sequence

constraint. Reformulating the problem by some mathematical programming tools, such as

mixed integer programming(MIP) and dynamic programming(DP), probably could provide

some alternative solution. For the particular problem of finding the optimal supply chain,


DP is more promising because most times the optimization problems formulated by MIP

are NP-hard. However, the following two factors could be the challenges if we want to use

DP to solve the problem. First, how the problem is formulated will definitely influence the

solution efficiency. For example, based on Figure 8, the optimal supply chain problem can

be formulated into the shortest path problem and DP can be used to find the solution. But

in that case, the number of nodes still increases exponentially when the number of modules

increases and therefore the computation burden does not reduce in this way. Therefore, how

to find a better way to formulate the problem so that the number of states will not increase

exponentially is critical for this optimization problem. Second, no matter how we formulate

the problem, finally it will be a complexity minimization problem and in each stage we

need to calculate the complexity incremental value based on the decision made at that state.

According to the assumption of DP, this complexity incremental value should be independent

from the future decisions. But recall that in our complexity definition, shown in Equation

(6), the complexity value of each node is related to the total number of arcs in the supply

chain, K, which is determined by the decision of current state and all future states. So in

our problem, this assumption is violated. However, we notice that 2m ≤ K ≤ 2m+ (m− 2),

where m is the number of nodes in the most upstream echelon. The lower bound, K = 2m,

is the case of non-modular supply chain where the number of arcs in the supply chain is

minimum, while the upper bound K = 2m + (m − 2), is the case of modular supply chain

with largest number of sub-assemblers, m− 2, where the number of arcs in the supply chain

is maximum. This relationship could help us to apply DP in solving the problem by setting

K to the upper or lower bound.


7 Conclusions

In this paper, a complexity measure of assembly supply chains is derived based on Shannon’s

information entropy. This complexity measure incorporates the detailed information of the

supply chain structure, the number of variants offered by each node in the supply chain, and

the mix ratios of the variants at each node. It is applied to solving the problem of finding the

optimal assembly supply chain given the number of variants offered at the final assembler

and the mix ratios of these variants. The optimal assembly supply chain configuration is

theoretically studied in the following two special scenarios: 1) there is one dominant variant

among all the variants offered by the final assembler, and 2) demand share are equal across all

variants at the final assembler. It is shown that in the scenario of one dominant variant, the

optimal assembly supply chain should be non-modular; but in the scenario of equal demand

shares, the modular supply chain is more beneficial than non-modular when the product

variety is high. For the general demand at the final assembler, a methodology is developed

to find the optimal supply chain if no assembly sequence constraint exists. An iterative

algorithm is proposed to generate all supply chain candidates. Then the methodology is

extended to solve the optimal supply chain problem with the existence of assembly sequence

constraints. A efficient method is developed to obtain all feasible supply chain candidates,

satisfying all the assembly constraints. The research of supply chain complexity generates

new insights on the influence of product variety on supply chains performance in mass

customization. It provides a model based analytic method, instead of empirical case study,

to the supply chain configuration selection problem in the presence of product variety. The

model and algorithms developed in this paper can assist in making decisions such as when


and how to implement a modular assembly supply chain and how much variety should be

economically offered.

Acknowledge

The authors gratefully acknowledge the financial support from the National Science Foun-

dation Engineering Research Center for Reconfigurable Manufacturing Systems (Award No.

EEC-9529125) and GOALI Project with General Motors (Award number: CMMI-0825438),

both at the University of Michigan.

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Appendix: Proofs

Proof of Proposition 1

Proof of (a)

The demand vector of the final assembler, node n, is qn := (qn1, qn2, . . . , qn,Vn). For

notation convenience, we assume the dominant variant is variant 1 and then the demand

share of the dominant variant is qn1.


For node i = 1, 2, . . . , n−1, we can divide the variants offered by the final assembler into

Vi disjoint subsets, each containing Vn/Vi variants of the final assembler, and the demand

share of the variants in each subset add up to the demand share of a variant at node i.

Without loss of generality, here we assume the demand share of variant 1 offered by node

i is obtained through summing the subset of variants, containing variant 1 from the final

assembler. Therefore, when the demand share of variant 1 at the final assembler increases

and approaches to 1, i.e., qn1 → 1, the demand share of variant 1 at node i = 1, 2, . . . , n− 1

also increases and approaches to 1, i.e., qi1 → 1. Because∑Vi

j=1 qij = 1 and qi1 → 1 for

i = 1, . . . , n, then the demand share of other variants at node i must approach 0, i.e.,

qij → 0, j = 2, . . . , Vi.

The complexity of an assembly supply chain defined in Equation (6) is C = −∑n

i=1 Li∑Vi

v=1qivK

log2qivK

, where Li is the number of supplies of node i and K is the number of arcs

in the supply chain, i.e, K =∑n

i=1 Li. With some algebra, it can be checked that the

complexity of Equation (6) can also be written as

C = log2K −1

K

n∑i=1

Li

Vi∑v=1

qiv log2 qiv. (8)

In the scenario of a dominant variant, we can rewrite the above equation as follows, where

qi1 → 1 and qij → 0, j = 2, . . . , Vi

C = log2K −1

K

n∑i=1

Li(qi1 log2 qi1 +

Vi∑v=2

qiv log2 qiv). (9)

As qi1 → 1, we have qi1 log2 qi1 = 1 · log2 1 = 0. But as qij → 0, j = 2, . . . , Vi, we have


qiv · log2 qiv, which is 0 · ∞ type limit and can be calculate through l’Hpital’s rule,

limqiv→0

qiv log2 qiv = limqiv→0

qiv log2 qiv1/qiv

= limqiv→0

(qiv log2 qiv)′

(1/qiv)′

= limqiv→0

−1/(ln 2 · qiv)1/q2iv

= limqiv→0

− qivln 2

= 0.

Then in the scenario of one dominant variant, the complexity of an assembly chain, rep-

resented as Equation (9), is C = log2K, where K is the number of total arcs in the supply

chain.

Proof of (b)

We want to make the decision of the supply chain configuration, given the number of

variants offered by the final assembler and the mix ratios of these variants. Following proof

(a), in the scenario of one dominant variant, the complexity of an assembly supply chain is

C = log2K, where K is the number of total arcs in the supply chain, including the arcs from

the virtual suppliers to the nodes in the most upstream echelon. Then the optimal assembly

supply chain should have the minimum number of arcs in the supply chain. According to the

definition of assembly supply chains, every node, except the virtual supplier and the final

assembler, only has one downstream node, then we can obtain the following relationship,

where n is the number of nodes in the supply chain and m is the number of nodes in the

most upstream echelon,

K = (n− 1) +m (10)

Since m is fixed by the number of module in the final product, in order to obtain the minimum

value of K in Equation (10), we should choose the assembly supply chain with minimum

number of nodes in the supply chain, which is the non-modular assembly supply chain. So


under the scenario of one dominant variant, the optimal assembly supply chain configuration

is non-modular.


Suppose the demand shares are equal across all the variants provided by the final as-

sembler, node n, i.e., qn = ( 1Vn, . . . , 1

Vn)1×Vn . By Equation (2), it is easily to verify that the

demand shares are also identical across all the variants at other nodes in the supply chain.

In addition, here we assume all the nodes in the most upstream echelon, node 1, . . . ,m,

provide the same number of variants, i.e, V1 = . . . = Vm = V . So the demand vector of the

nodes in the most upstream echelon takes the form of qi = ( 1V, . . . , 1

V)1×V , i = 1, . . .m. Since

we assume that a node can assemble any combination of the components from its upstream

suppliers, then Vn =∏m

i=1 Vi = V m.

We want to make the decision of the supply chain configuration, given the number of

variants offered by the final assembler and the mix ratios of these variants. All the supply

chains, which provide the same number of variants at the final assembler and have the same

mix ratios of these variants, must have the same number of nodes in the most upstream

echelon, m. Next, we will prove that when V is big enough, there is at least one modular

assembly supply chain, which has less complexity value than non-modular assembly supply

chain.

Let us consider the following two assembly supply chains, which has the same demand

vector of the final assembler, qn = ( 1Vn, . . . , 1

Vn)1×Vn = ( 1

V m , . . . ,1

V m )1×V m . One is non-

modular assembly supply chain in Figure 12(a) and another is modular assembly supply chain


Figure 12: One non-modular assembly supply chain and one modular assembly with only

one intermediate sub-assembler

in Figure 12(b), which has one sub-assembler in the middle echelon, assembling components

from node 1, . . . ,m−1. The demand vector of this sub-assembler in modular assembly supply

chain is qn−1 = ( 1V m−1 , . . . ,

1V m−1 )1×V m−1 . The complexity of an assembly supply chain can

be calculated through Equation (8), i.e., C = log2K − 1K

∑ni=1 Li

∑Vi

v=1 qiv log2 qiv.

For non-modular assembly supply chain, we substitute qi = ( 1V, . . . , 1

V)1×V , Li = 1, i =

1, . . .m, qn = ( 1V m , . . . ,

1V m )1×V m , Ln = m and K = 2m back to Equation (8) and get the

complexity of the non-modular assembly supply chain in Figure 12(a),

Cnon−mod = log2 2m+m+ 1

2log2 V

Following the same argument, by substituting qi = ( 1V, . . . , 1

V)1×V , Li = 1, i = 1, . . .m,

qn−1 = ( 1V m−1 , . . . ,

1V m−1 )1×V m−1 , Ln−1 = m − 1, qn = ( 1

V m , . . . ,1

V m )1×V m , Ln = 2 and K =

2m + 1 back to Equation (8), we can get the complexity of the modular assembly supply

chain in Figure 12(b),

Cmod = log2(2m+ 1) +m2 +m+ 1

2m+ 1log2 V

Then the complexity difference between modular and non-modular assembly supply chain

in Figure 12 is,

Cmod − Cnon−mod = log2

2m+ 1

2m− m− 1

2(2m+ 1)log2 V (11)


Since m is the number of nodes in the most upstream echelon of the assembly supply

chain, m must a integer greater than 1, i.e., m > 1, a ∈ Z, which results in 2m+12m

> 1

and m−12(2m+1)

> 0. For a fixed m, define the following two constants, A = log22m+12m

> 0

and B = m−12(2m+1)

> 0. Then the difference between non-modular and modular assem-

bly supply chain in Equation (11), can be rewritten as Cmod − Cnon−mod = A − B log2 V ,

which is a decreasing function of V. There must be a threshold, t, so that when V ≥ t,

Cmod − Cnon−mod < 0, which makes the modular assembly chain more preferable. So when

V is big enough, i.e., V ≥ t, we find at least one modular assembly supply chain, shown

in Figure 12(b), has less complexity value than the corresponding non-modular assembly

supply chain, shown in Figure 12(a).


Suppose the demand shares are equal across all the variants provided by the final as-

sembler, node n, i.e., qn = ( 1Vn, . . . , 1

Vn)1×Vn . By Equation (2), it is easily to see that the

demand shares are also identical across all the variants at other nodes in the supply chain. In

addition, since all the nodes in the most upstream echelon, node 1, . . . ,m, provide the same

number of variants, i.e, V1 = . . . = Vm = V , then we have qi = ( 1V, . . . , 1

V)1×V , i = 1, . . .m

and Vn =∏m

i=1 Vi = V m.

Proof of (a)

From the proposition statement, it is easy to see that m, can be written as the form of

the product of two positive integers, i.e., m = c × d, c, d ∈ Z+. Now consider the following


two supply chains: Both of them satisfy the above assumptions and have one intermediate

echelon, in which there are several sub-assemblers. Supply chain 1 has c intermediate sub-

assemblers, each of which has d suppliers from the most upstream echelon and then has

the demand vector as q(1)i = ( 1

V d , . . . ,1V d )1×V d , i = m + 1, . . .m + c. Supply chain 2 has d

intermediate sub-assemblers, each of which has c suppliers from the most upstream echelon

and then has q(2)i = ( 1

V c , . . . ,1V c )1×V c , i = m+ 1, . . .m+ d.

Recall in the proof of Proposition 1, the complexity of an assembly supply chain can

be calculated through Equation (8), i.e., C = log2K − 1K

∑ni=1 Li

∑Vi

v=1 qiv log2 qiv. For

supply chain 1, there are total 2m + c arcs in the supply chain, i.e., K(1) = 2m + c. We

substitute q(1)i = ( 1

V, . . . , 1

V)1×V , L

(1)i = 1, i = 1, . . .m and q

(1)i = ( 1

V d , . . . ,1V d )1×V d , L

(1)i = d,

i = m + 1, . . .m + c and q(1)n = ( 1

V m , . . . ,1

V m )1×V m , L(1)n = c and K(1) = 2m + c back to

Equation (8) to get the complexity of the supply chain 1,

C(1) = log2(2m+ c) +m+md+mc

2m+ clog2 V

Similarly, for supply chain 2, there are total 2m+d arcs in the supply chain, i.e., K(2) = 2m+

d. We substitute q(2)i = ( 1

V, . . . , 1

V)1×V , L

(2)i = 1, i = 1, . . .m and q

(2)i = ( 1

V c , . . . ,1V c )1×V c ,

L(2)i = c, i = m + 1, . . .m + d and q

(2)n = ( 1

V m , . . . ,1

V m )1×V m , L(2)n = d and K(2) = 2m + d

back to Equation (8) to get the complexity of the supply chain 2,

C(2) = log2(2m+ d) +m+md+mc

2m+ dlog2 V

Then the complexity difference between these two supply chains is,

C(1) − C(2) = log2

2m+ c

2m+ d+

m+md+mc

(2m+ c)(2m+ d)(d− c) log2 V

Let A = log22m+c2m+d

and B = m+md+mc(2m+c)(2m+d)

(d− c). If c ≤ d, then A ≤ 0 and B ≥ 0 and we


can rewrite the complexity difference as C(1) −C(2) = A+B · log2 V , which is an increasing

function of V . Then there must be a threshold, η, so that when V ≥ η, C(1)−C(2) ≥ 0. We

can conclude that when c ≤ d and V ≥ η, C(1) ≥ C(2), which means the modular assembly

supply chain with larger number of intermediate sub-assemblers has less complexity value.

Proof of (b)

Now consider the following two supply chains: Both of them have one intermediate

echelon, where there are c sub-assemblers. In supply chain 1, all intermediate sub-assemblers

have the same number of suppliers from the most upstream echelon, d, i.e., L(1)m+i = d, i =

1, . . . , c, and∑c

i=1 L(1)m+i = cd = m. In supply chin 2, each intermediate sub-assembler has

different number of suppliers from the most upstream echelon. But since one node can only

supply one downstream node, then for supply chain 2, the sum of the number of suppliers

for all the these c sub-assemblers is still cd, i.e,∑c

i=1 L(2)m+i = cd = m. We can rewrite

the number of suppliers for these c sub-assemblers in supply chain 2 in the following form,

L(2)m+i = d+ ei, ei ∈ Z,

∑ci=1 ei = 0, i = 1, . . . , c.

Each sub-assembler in supply chain 1 has d suppliers from the most upstream echelon,

resulting the following demand vectors: q(1)m+i = ( 1

V d , . . . ,1V d )1×V d , i = 1, . . . , c. Each sub-

assembler in supply chain 2 has d + ei suppliers from the most upstream echelon, which

results in the following demand vector: q(2)m+i = ( 1

V d+ei, . . . , 1

V d+ei)1×V d+ei , i = 1, . . . c.

The complexity of an assembly supply chain can be calculated through Equation (8), i.e.,

C = log2K − 1K

∑ni=1 Li

∑Vi

v=1 qiv log2 qiv. We substitute q(1)i = ( 1

V, . . . , 1

V)1×V , L

(1)i = 1, i =

1, . . .m and q(1)m+i = ( 1

V d , . . . ,1V d )1×V d , L

(1)m+i = d, i = 1, . . . c and q

(1)n = ( 1

V m , . . . ,1

V m )1×V m , L(1)n =


c and K(1) = 2m+ c back to Equation (8) to get the complexity of the supply chain 1,

C(1) = log2(2m+ c) +m+ cd2 +mc

2m+ clog2 V

Similarly, we substitute q(2)i = ( 1

V, . . . , 1

V)1×V , L

(2)i = 1, i = 1, . . .m and q

(2)m+i =

( 1V d+ei

, . . . , 1V d+ei

)1×V d+ei , L(2)m+i = d + ei, i = 1, . . . , c and q

(2)n = ( 1

V m , . . . ,1

V m )1×V m , L(2)n = c

and K(2) = 2m+ c back to Equation (8) to get the complexity of the supply chain 2,

C(2) = log2(2m+ c) +m+ c

∑ci=1(d+ ei)

2 +mc

2m+ clog2 V

The complexity difference between supply chain 1 and supply chain 2 is,

C(1) − C(2) =c log2 V

2m+ c

[d2 −

c∑i=1

(d+ ei)2

]

=c log2 V

2m+ c

[d2 −

c∑i=1

(d2 + 2dei + e2i )

]

= −c log2 V

2m+ c

[2d

c∑i=1

ei +c∑

i=1

e2i

]

Recall that∑ei = 0, then C(1) − C(2) = − c log2 V

2m+c

∑ci=1 e

2i . Since V ≥ 1, m, c ∈ Z+ and∑c

i=1 e2i ≥ 0, we get C(1)−C(2) ≤ 0. So it is concluded that in the scenario of equal demand

shares, the complexity of supply chain 1 is less than supply chain 2.

List of Figures

1 Manufacturers move from non-modular assembly supply chains to modular

assembly supply chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 A general assembly supply chain and relationships of variants’ demand share 9

3 The example used to illustrate the calculation of assembly supply chain com-

plexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13


4 Three modular assembly supply chains with 6 nodes in the most upstream

echelon and one intermediate echelon . . . . . . . . . . . . . . . . . . . . . . 19

5 The example used to illustrate the methodology of finding the optimal assem-

bly supply chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 There are five configurations to connect the four nodes in the most upstream

echelon to the final assembler . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7 There are more than one supply chain candidates for the same configuration 23

8 Iterative decomposition algorithm to generate all possible supply chain can-

didates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

9 A set of assembly sequence constraints . . . . . . . . . . . . . . . . . . . . . 27

10 Feasibility check of intermediate supply chain candidates after each decompo-

sition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

11 Feasibility check criteria for an intermediate assembly supply chain . . . . . 30

12 One non-modular assembly supply chain and one modular assembly with only

one intermediate sub-assembler . . . . . . . . . . . . . . . . . . . . . . . . . 41


a complexity model for assembly supply chains and its...

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